Mathematical mosaics is a new term for me. I have always called the mosaics types of tessellations, a tiling made by combining polygons to create repeatable patterns to cover a plane. Through this lesson, I learned mathematical mosaic refers to a tessellation that has only one numerical representation. It can be made of entirely of one regular polygon creating a regular mathematical mosaic or a combination of several regular polygons to create a semi-regular mathematical mosaic.
Identifying polygons that surround a point is the first step in the process. Exercises 1-4 on page 271 demonstrate a variety of patterns: 6-6-6, 4-8-8, 3-3-3-3-6, and 4-6-4-12.
The key to creating a mathematical mosaic that successfully fills a plane is the sum of the angles surrounding any given point in the mosaic. The sum must be 3600. In exercise 5 on page 272 the sum of the angles is 3630 (108 + 120 + 135). Sometimes, when looking at just a few polygons and one or two points, a configuration may look plausible, like exercise 6-9. A 5-5-10 looks like it would create a mosaic because the sum of the angles is 3600 (108 +108 + 144) but when putting several units together, not all points create 3600. Point C in the diagram has three pentagons adding to 3240 leaving a gap. Although there are tessellations having two numerical representations (called demi-regular tessellations), there are never any gaps and the patterns always continue.
Questions I would ask my students:
1. Create a semi-regular mathematical mosaic using two or three regular polygons of your choice. Name the numerical representation of your mosaic.
2. Using only triangles and hexagons, create a tessellation. Explain why or why not your tessellation is a mathematical mosaic.
3. Study these mathematical mosaics. How are they alike and how are they different? (Images from http://www.coolmath4kids.com/tesspag1.html)
